The Hadamard product and the free convolutions

Article Type

Research Article

Publication Title

Statistics and Probability Letters

Abstract

It is shown that if a probability measure ν is supported on a closed subset of (0,∞), that is, its support is bounded away from zero, then the free multiplicative convolution of ν and the semicircle law is absolutely continuous with respect to the Lebesgue measure. For the proof, a result concerning the Hadamard product of a deterministic matrix and a scaled Wigner matrix is proved and subsequently used. As a byproduct, a result, showing that the limiting spectral distribution of the Hadamard product is same as that of a symmetric random matrix with entries from a mean zero stationary Gaussian process, is obtained.

First Page

150

Last Page

157

DOI

10.1016/j.spl.2017.04.009

Publication Date

8-1-2017

Comments

Open Access, Green

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